Detailed contributions

sect:ctb-1-2-1-ref-ind-and-col

If both the main delay line and differential delay line are filled with air, then the measured distances within the VLTI must be converted to OPD at the observing wavelength for each star. The observing wavelength for each star is the centroid of the correlated flux detected for that star. If the observing wavelength of the primary star is $\lambda_{ps}$, the observing wavelength of the secondary star is $\lambda_{ses}$ and the metrology wavelength is $\lambda_{m}$, then the $\Delta$OPD measured at the laser wavelength $\Delta D_{l}$ is given by:

$$\Delta D_{l} = \frac{\left (D+\Delta D\right )n_{\lambda_{ses}}}{n_{\lambda_{m}}} - \frac{D n_{\lambda_{ps}}}{n_{\lambda_{m}}}$$ (19)

$$\Delta D_{l} \simeq \left (D+\Delta D\right )\left (1+n_{\lambda_{ses}}-n_{\lambda_{m}}\right ) - D \left (1+n_{\lambda_{ps}}-n_{\lambda_{m}}\right )$$ (20)

$$= \Delta D + \Delta D \left (n_{\lambda_{ses}} -n_{\lambda_{m}}\right) + D \left (n_{\lambda_{ses}}-n_{\lambda_{ps}} \right )$$ (21)

where $D$ is the OPD corrected in the main delay line, and $\Delta D$ is the OPD corrected in the DDL. The second term on the right hand side of Equation 22 corresponds to the effect of the air in the DDL, while the third term correponds to the effect of the air in the main delay line. The approximation in Equation 21 is fully valid for this error analysis due to the low refractivity of air.

We wish to measure $\Delta D$:

$$\Delta D = \Delta D_{l}-\Delta D \left (n_{\lambda_{ses}} -n... ...\right) - D \left (n_{\lambda_{ses}}-n_{\lambda_{ps}} \right )$$ (22)

The conversion from $\Delta D_{l}$ would be done using the best estimates of $n_{\lambda_{ses}}$ and $n_{\lambda_{ps}}$ available. The error $\epsilon \left(\Delta D\right )$ in the differential OPD measurement $\Delta D$ can be separated into a number of principle terms:

$$\epsilon \left(\Delta D\right ) = \epsilon \left(\Delta D_{l... ...- D \epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{ps}} \right )$$ (23)

where

$$\epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )$$ (24)

is the error in estimation of $n_{\lambda_{ses}}-n_{\lambda_{m}}$,

$$\epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )$$ (25)

is the error in estimation of $n_{\lambda_{ses}}-n_{\lambda_{ps}}$ and

$$\epsilon \left(\Delta D_{l}\right )$$ (26)

is the (hopefully insignificant) error in the ability of the metrology system to count laser wavelengths.

The error terms in Equations 25 and 26 can be further broken down by taking partial derivatives. I will use the following approach:

$$Z = XY$$ (27)

$$\Delta Z = \Delta X\frac{\partial\left(XY\right )}{\partial X}+\Delta Y\frac{\partial\left(XY\right )}{\partial Y}$$ (28)

Applying this approach to Equation 25 yields:

$$\epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right ) = \left \{ \begin{array}{l} \epsilon \left(\rho_{a}\right ) \frac{\... ...bda}\right ) \left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right ) \end{array}\right.$$ (29)

$$\epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right ) = \left \{ \begin{array}{l} \epsilon \left(\rho_{a}\right ) \frac{\... ...da}\right ) \left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right ) \end{array}\right.$$ (30)

where $\rho_{a}$ is the air density and $\rho_{w}$ is the water vapour density in the VLTI. $\epsilon \left ( \frac{\Delta\left(n_{\lambda}\right )}{\Delta \lambda}\right )$ is the uncertainty in the model for dispersion of the air and water vapour in the VLTI between two wavelengths.

Typical values of the components making up these error terms are listed in Table 1.

表 1: Individual components of the error terms listed in Equation 30

Term	Typical value	Units
$\epsilon \left(\rho_{a}\right )$	$0.005$	fractional error
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )}{\partial \rho_{a}}$	$3\times10^{-9}$	$\frac{\mbox{fractional change}}{\mbox{fractional change}}$
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \rho_{a}}$	$1\times10^{-11}$	$\frac{\mbox{fractional change}}{\mbox{fractional change}}$
$\epsilon \left(\rho_{w}\right )$	$0.0005$	fractional error
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )}{\partial \rho_{w}}$	$1\times10^{-8}$	$\frac{\mbox{fractional change}}{\mbox{fractional change}}$
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \rho_{w}}$	$3\times10^{-11}$	$\frac{\mbox{fractional change}}{\mbox{fractional change}}$
$\epsilon \left(\lambda_{ses}\right )$	$2$	$\mbox{nm}$
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )}{\partial \lambda_{ses}}$	$4\times10^{-10}$	$\frac{\mbox{fractional change}}{\mbox{nm}}$
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \lambda_{ses}}$	$4\times10^{-10}$	$\frac{\mbox{fractional change}}{\mbox{nm}}$
$\epsilon \left(\lambda_{ps}\right )$	$2$	$\mbox{nm}$
$\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \lambda_{ps}}$	$4\times10^{-10}$	$\frac{\mbox{fractional change}}{\mbox{nm}}$
$\epsilon \left ( \frac{\Delta\left(n_{\lambda}\right )}{\Delta \lambda}\right )$	Unknown in the near infrared

From Table 1 we can calculate approximate magnitudes for the error terms listed in Equation 30 for the air path in the DDL

$$\epsilon \left(\rho_{a}\right )\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )}{\partial \rho_{a}} \simeq 2\times10^{-11}$$ (31)

$$\epsilon \left(\rho_{w}\right )\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )}{\partial \rho_{w}} \simeq 5\times10^{-12}$$ (32)

$$\epsilon \left(\lambda_{ses}\right ) \frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right )}{\partial \lambda_{ses}} \simeq 8\times10^{-10}$$ (33)

with an additional term $\epsilon \left ( \frac{\Delta\left(n_{\lambda}\right )}{\Delta \lambda}\right )$ from the dispersive properties of air, which is currently unknown.

If a reliable model for the refractive index of air can be constructed, then it is clear that the error in the colour of the observing band for the secondary star will dominate over the contribution from air in the DDL, giving an total error contribution of (summing errors in quadrature):

$$\epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{m}}\right ) \simeq 8\times10^{-10}$$ (34)

in units of metres of OPD error per metre air path in the DDL. This corresponds to $0.08\mbox{ nm}$ error for $10\mbox{ cm}$ of air path in the DDL.

Equations 32 to 34 are listed alongside a verbal description in Table 2.

表 2: Error in differential OPD per unit length of air path in the DDL introduced by uncertainties in the observing conditions

Equation		m $\Delta OPD$
number	Description	per m air
32	The effect of $0.5\%$ uncertainty in the density of the air in the DDL, due to uncertainties in temperature and pressure	$2\times10^{-11}$
33	The effect of $0.2\%$ uncertainty in the density of the water vapour in the DDL, due to uncertainties in relative humidity, temperature and pressure	$5\times10^{-12}$
34	The effect of a $2\mbox{ nm}$ uncertainty in the centroid wavelength of the correlated flux from secondary star	$8\times10^{-10}$

From Table 1 we can also calculate approximate magnitudes for the error terms listed in Equation 31 for the air path in the main delay line:

$$\epsilon \left(\rho_{a}\right )\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \rho_{a}} \simeq 1\times10^{-13}$$ (35)

$$\epsilon \left(\rho_{w}\right )\frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \rho_{w}} \simeq 3\times10^{-14}$$ (36)

$$\epsilon \left(\lambda_{ses}\right ) \frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \lambda_{ses}} \simeq 8\times10^{-10}$$ (37)

$$\epsilon \left(\lambda_{ps}\right ) \frac{\partial\left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right )}{\partial \lambda_{ps}} \simeq 8\times10^{-10}$$ (38)

with an additional term $\epsilon \left ( \frac{\Delta\left(n_{\lambda}\right )}{\Delta \lambda}\right )$ from the dispersive properties of air, which is currently unknown.

If the refractive indices of air and water vapour can be measured with sufficient reliability, it is clear that the error in the colour of the observing band for the primary and secondary stars will dominate the contribution from air in the main delay line, giving a total error contribution of (summing errors in quadrature):

$$\epsilon \left(n_{\lambda_{ses}}-n_{\lambda_{ps}}\right ) = ... ..._{\lambda_{ses}}-n_{\lambda_{m}}\right ) \simeq 1\times10^{-9}$$ (39)

in units of metres of OPD error per metre of air path compensating vacuum path with the main delay lines. This corresponds to $100\mbox{ nm}$ differential OPD error for $100\mbox{ m}$ of air path in the main delay line.

Equations 36 to 39 are listed alongside a verbal description in Table 3.

表 3: Error in differential OPD introduced by uncertainties in the observing conditions per unit length of air path in the main delay lines and AT ducts which is balancing vacuum path (distance of the main delay line from that of equal OPD to the two telescopes)

Equation		m $\Delta OPD$
number	Description	per m air
36	The effect of $0.5\%$ uncertainty in the density of the air in the main delay line, due to uncertainties in temperature and pressure	$1\times10^{-12}$
37	The effect of $0.2\%$ uncertainty in the density of the water vapour in the main delay line, due to uncertainties in relative humidity, temperature and pressure	$3\times10^{-14}$
38	The effect of a $2\mbox{ nm}$ uncertainty in the centroid wavelength of the correlated flux from secondary star	$8\times10^{-10}$
39	The effect of a $2\mbox{ nm}$ uncertainty in the centroid wavelength of the correlated flux from primary star	$8\times10^{-10}$